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Flight-Calc 2018 Documentation
W. B Garner, February 2018
Introduction
Background A usual way of estimating in-flight performance of electric powered model airplane is to use one of the on-line calculators, guessing about size of propellers, motors & related items, and then iterating the inputs until a ‘satisfactory’ result is obtained. While this method works it is cumbersome and doesn’t explicitly set performance goals as part of the input process. The Flight-Calc program starts with the desired flight performance criteria and builds from there.
Scope The program takes as input the characteristics of the plane to be analyzed and lets the user select a
particular flight condition for synthesis purposes. The user then selects a trial propeller size and the
program provides an estimate of the motor characteristics necessary to meet the flight criteria. This is
followed by a program that takes as input the propeller and battery size selected and a specific set of
motor specifications supplied by the user. The results are then presented in the form of a table and a
graph.
Structure The program is run in Excel and consists of five user sheets. There are other sheets that are internal to
the program, not intended for the user.
The first sheet, labeled “Notes”, provides instructions on how to run the program as well as describing
its limitations.
The second sheet, labeled “Airplane Inputs”, takes in user supplied information about the airplane’s
physical characteristics. Its output is a flight performance graph from which the user may select a
specific operating point to use in the following analysis.
The third sheet, labeled “Prop & Motor Selection”, takes as input the selected flight operating point &
user selected propeller sizes. The program then computes the required propeller rpm and input power
necessary to meet the flight conditions. Estimates of the required motor characteristics are then made
as a function of several battery voltages. The user can change the propeller size to see how the motor
characteristics must change to meet the flight criteria. The user then selects the desired battery voltage
and motor parameters for use in the fourth sheet.
The fourth sheet, labeled “Specific Prop-Motor Results”, has as input the results of sheet three and user
supplied motor specifications. The output is a Table listing the flight performance as a function of
airspeed as well as the motor and battery operating values.
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The fifth sheet, labeled “Results Summary”, is the final sheet, presenting in one place all of the
assumptions and results. The primary results are the Table from sheet four and a graph showing how
the analysis results relate to the original flight performance graph.
Airplane Input Sheet
Inputs The inputs for this sheet are wing area, wing span, weight, fuselage area, tail area and altitude. In
addition the user selects a wing type and a fuselage configuration that most closely matches the plane
from lists. The program estimates the drag as a function of airspeed, climb angle and airspeed and
estimates the thrust required to meet each flight condition.
Calculation Model Figure 1 is a diagram illustrating the forces acting on an airplane in climbing flight.
The resulting equilibrium forces are given by the following equations.
L = W*cos (Ɵ) T = D + W* sin (Ɵ) Where: L is lift W is weight T is thrust Ɵ is the angle between horizontal and the velocity vector V is the air speed The associated rate of climb, ROC, is given by:
ROC = (T*V – D*V)/W, ft/sec
And the angle of climb, AOC, is given by:
AOC=arcsine ((T-D)/W), radians
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The rate of climb or descent is zero when T =D, positive when T > D and negative when T < D. The ROC is
inversely proportional to the weight, so an increase in weight results in a decrease in climb rate. ROC
increases with a decrease in weight.
In the present case the airspeed, climb angle and weight are given and it is desired to find the required
thrust and the drag associated with the airspeed.
T = D + W*sin(AOC)
When the climb angle is zero the second term is zero and when the climb angle is 90 degrees the
required thrust is equal to the drag plus the weight. These two values of climb angle set the boundaries
of powered flight.
The program analyzes the drag as composed of four components. They are wing profile drag, wing
induced drag, tail drag and fuselage drag. The wing profile drag is modeled as a function of lift
coefficient, CL. The wing induced drag is modeled taking into account the effect of climb angle on the
resulting lift coefficient. It goes to zero when the climb angle is 90degrees, as the wing is then at zero lift
condition.
Formulas:
Cl = 𝑊
𝑟ℎ𝑜∗𝑆𝑤∗𝑉2∗cos(𝐴𝑂𝐶)
Cdprofile = F(Cl) F(Cl) is a polynomial with the variable being Cl
Profile Drag: Dprofile =𝐶𝑑𝑝𝑟𝑜𝑓𝑖𝑙𝑒 ∗ 𝑟ℎ𝑜 ∗ 𝑆𝑤 ∗ 𝑉2
Induced Drag: Dinduced =0.318∗𝑊2∗(1+𝑑𝑒𝑙𝑡𝑎)∗cos(𝐴𝑂𝐶)
𝑟ℎ𝑜∗𝑠𝑝𝑎𝑛2∗𝑉^2
Tail Drag: Dtail =𝐶𝑑𝑡𝑎𝑖𝑙 ∗ 𝑟ℎ𝑜 ∗ 𝑆𝑡 ∗ 𝑉2
Fuselage Drag: Dfus =𝐶𝑑𝑓𝑢𝑠 ∗ 𝑟ℎ𝑜 ∗ 𝑆𝑓 ∗ 𝑉2
Total Drag: Drag = Dprofile + Dinduced + Dtail + Dfuselage
Where:
rho is air density Sw is wing area V is airspeed W is weight Cl is lift coefficient span is wing span delta is a wing planform adjustment, = .05 Cdtail is tail drag coefficient (0.03) St is total tail area
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Cdfus is fuselage drag coefficient Sf is maximum fuselage cross-section area The program calculates the required equilibrium thrust for each airspeed- climb angle combination and
displays the results in the form of a graph.
Results The user selects an operating point of interest and enters the corresponding required thrust and
airspeed into an input box automatically transferred to the next sheet.
Example
This example illustrates how the program is used for selecting a flight operating point. The plane is an
electric powered stick-like one with the listed characteristics. The wing type selection number 2 is
chosen as the wing has a thick symmetrical cross section. The fuselage selection number 4 indicates that
the fuselage is shaped like a typical stick airplane with fixed landing gear.
Name Value Units
wing area 600 in^2
wing span 50 in
weight 32 oz
tail area 120 in^2
fuselage max cross section area 9 in^2
altitude 6800 feet
wing type selection number 2
fuselage type selection number 4
Drag Multiplier 1
Wing Type Selection List
Select # Description Typical Airfoil
1 Thin Symmetrical Eppler 478
2 Thick Symmetrical NACA 010
3 Thin Asymmetrical RG-14
4 Thick Asymmetrical Clark Y
5 Sail Plane S3014
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The resulting flight performance graph is shown in Figure 2.
The graph displays the required equilibrium thrust as a function of airspeed and climb angle. The top
curve is the case where the climb angle is 90 degrees (straight up) and where there is no wing lift. The
thrust must equal the plane weight plus the drag as the airspeed increases. Note that this curve
increases as the square of the airspeed, as do all of the others except near stall where the lower curves
terminate.
The lowest curve displays the thrust required at zero climb angle, or level flight. The plane stalls at about
15 MPH, the thrust decreases then increases as the airspeed increases. Other climb angle values lie
between the zero and ninety degree curves. Hence the plane operating range is bounded by these two
curves.
Fuselage Types
Select # Description Cdfus
1 War Bird, Fixed Gear 0.75
2 War Bird, Retracts 0.5
3 Trainer, Boxy, Cabin 1.5
4 Stick, fixed gear 1.2
5 Sailplane, no gear 0.25
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In this example, an operating point is selected at 40 mph and zero climb-angle. The required thrust is 2.6
pounds. Note that this thrust value should allow the plane to go vertical at about 20 mph or less.
The values of 2.6 pounds thrust and 40 mph are selected and passed to the next sheet.
Prop & Motor Selection
Summary This section is the central tradeoff part of the analysis. The user selects a propeller size, diameter and
pitch, from the list of allowed values. The program then calculates the propeller rpm and input power
required to meet the required thrust and airspeed values. Next the program estimates the required
motor current, power and Kv needed for each of four battery voltages. The user then selects the battery
voltage and corresponding motor properties for further detailed investigation.
Inputs The inputs are propeller diameter and pitch selected from a list of allowed APCE propellers ranging in
size from 8 to 14 inches. The thrust and power coefficients associated with these propellers were
derived from wind tunnel data with interpolation for sizes not included in the wind tunnel data. The
program will return an error if a size is not included in the list.
The input also includes a box for entering a Watts per gram value to help identify the weight of a
suitable motor.
Calculation Method The program then computes the value of rpm and input power required to match the required thrust for
the selected propeller.
Tcalc = 𝐶𝑡(𝐽) ∗ 𝑟ℎ𝑜 ∗ (𝑟𝑝𝑚
60)2∗ (
𝐷𝑖𝑎𝑚𝑒𝑡𝑒𝑟
12) ∗ 𝑉2
J = 1056∗𝑉
𝑟𝑝𝑚∗𝐷 Where J is the advance ratio
Since rpm is not known, RPM is changed in increments until the value of Tcalc matches the value of
required T. The value of the required propeller input power is then calculated. The result is presented in
the form of a table. In this case the propeller size is 10x7.
Match Results
T Lbs Prop Watts Rpm
Pbatt Watts
Weight, grm
2.59 290 10763 341 114
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The results show that a thrust value of 2.59 pounds is achieved at a propeller input power of 290 Watts
and an rpm = 10,763. The estimated battery power is 341 Watts and the motor should weight about 114
grams.
The next part of the program takes these results and applies them to a generic electric motor model to
estimate the required Kv as a function of battery voltage.
A simple motor model is implemented, consisting of a battery in series with a battery resistance, ESC
resistance and a motor resistor and opposed by the motor EMF voltage. The total current is equal to the
battery power previously calculated divided by the battery voltage. The battery is assumed to be able to
support the full current for 6 minutes, its C rating is 35, and its voltage is a multiple of 4 Volts.
Rbattery = 0.037∗𝐸𝑏𝑎𝑡𝑡𝑒𝑟𝑦
𝐼𝑏𝑎𝑡𝑡𝑒𝑟𝑦 + .005
The ESC resistance is modeled as: Resc = = .281 ∗ 𝐼𝑏𝑎𝑡𝑡𝑒𝑟𝑦−1.24
The motor resistance is modeled as: Rm =𝑎 ∗ 𝑃𝑏𝑎𝑡𝑡𝑒𝑟𝑦−𝑏 where a = 6.9 and b = .843
Kv = 𝑟𝑝𝑚
𝐸𝑏𝑎𝑡𝑡𝑒𝑟𝑦−𝑃𝑏𝑎𝑡𝑡𝑒𝑟𝑦
𝐸𝑏𝑎𝑡𝑡𝑒𝑟𝑦∗𝑅𝑡𝑜𝑡𝑎𝑙
The results for this example are listed in the following table.
Motor Performance Options
Ebattery Ibattery Kv
8.0 44 2176
12.0 29 1118
16.0 22 776
20.0 17 600
24.0 15 490
This table shows the tradeoff between battery voltage, current and Kv where each case matches the
thrust, prop input power and rpm requirements.
The next stage is for the user to select a battery voltage and input detailed actual motor specifications
using the selected propeller size. This example selects a voltage of 12 Volts with a corresponding Kv of
1118.
The selected values are used to estimate the thrust and other performance characteristics as a function
of airspeed as illustrated in the following table. These results are overlaid on the graph generated for
the Airplane Input section as shown in the following figure.
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Results
Vmph Tprop Iamps RPM Pout Pbattery
0 3.4 33.4 10643 289 401
10 3.4 31.9 10763 293 383
20 3.1 33.4 10643 291 401
30 2.9 33.4 10643 293 401
40 2.6 31.9 10763 290 383
50 2.2 30.5 10884 266 366
60 1.8 26.1 11246 245 314
70 1.4 21.8 11608 208 261
The graph illustrates how the thrust varies with airspeed and how it compares to the required thrust at a
given airspeed and climb angle.
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Specific Prop-Motor Results
Inputs A suitable motor will have a Kv of about 1000, have a power rating greater than 350 Watts, a current
rating greater than 30 Amps and a weight of about 114 grams. The next table lists the specifications for
an actual motor approximately meeting or exceeding these requirements. The weight is 134 grams.
Motor Parameters Kv 1000 rpm/volts
Rm 0.032 Ohms
Io 1 Amps
Vo 10 Volts
Imax 45 Amps
Pmax 670 Watts
Calculation Method There is no closed form method for matching the motor to the prop for a given set of required propeller
inputs. The method used is to increment the total current in steps from a low value to a high value,
calculate the resulting motor rpm and power. Then choose airspeed and calculate the prop input power
and thrust given the motor rpm for each current increment. Compare the motor output power to the
propeller input power until they match. At that current and rpm the solution is found. The airspeed is
then incremented to the next value and the process repeated, resulting in a table listing the results as
follows.
Formulas:
EMF = Ebattery – Itotal*Rtotal
RPM = Kv * EMF
Imotor = 𝐼𝑡𝑜𝑡𝑎𝑙 −𝐼𝑜
𝑉𝑜∗ 𝐸𝑀𝐹
Pout = 𝐼𝑚𝑜𝑡𝑜𝑟∗𝑟𝑝𝑚
𝐾𝑣
Pbattery = Itotal * Ebattery
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Results Summary The final sheet summarizes all of the inputs and assumptions used in the analysis and present the results
in the form of tables and a graph. The graph is similar to the one from the Prop and Motor Section
sheet.
The values of specific motor thrust are slightly lower than for the estimated thrust conditions, due in
part to the lower than estimated Kv.
Performance Results
Vmph Thrust, lbs I, amps RPM Pout, W Pbattery, W
0 3.2 31.9 10352 320 383
10 3.1 31.9 10352 320 383
20 3.0 31.9 10352 320 383
30 2.7 31.9 10352 320 383
40 2.5 27.6 10577 281 331
50 2.0 27.6 10577 281 331
60 1.6 23.2 10801 239 279
70 1.1 17.4 11101 181 209
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Comments The program uses the foot – pound – second units of measure. The Airplane Input section contains a
converter from metric to this system.
Drag coefficients are the parameters most subject to error in the Airplane Input section. The Airplane
Input sheet has a box labeled “Drag Multiplier”. Any number inserted into it will multiply the total
computed drag by that amount.
The motor model is a simplified version of an actual motor model. It assumes that the idle current for all
motors is equal to 1 Amp at 10 Volts. The battery, ESC and motor estimated resistances may be
different from actual devices.
Kv is the most sensitive parameter in selecting or analyzing a motor. The thrust varies as the square of
Kv. For instance, a difference of 5% will change the thrust by 1.05^2 = 1.10, or 10 %. The power is
proportional to the cube of the Kv so a 5% difference will result in a 1.05^3 =1.16, or 16% difference in
power. Actual motors tend to come in discrete Kv values, so matching the estimated Kv with a real
motor may be a challenge.
Appendices:
Airfoil Drag Coefficients
These drag coefficients were generated using the Profili XT program.
Thick Symmetrical: NACA 010
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Thin Symmetrical: Eppler 478 Thin Asymmetrical: RG-14 Thick Asymmetrical: Clark Y Sail Plane: S3014
Fuselage Drag Coefficients The fuselage drag coefficients were derived from Andy Lennon, “Basics of RC Model Aircraft Design”
Model Airplane News, 2002, Chapter 12.
The coefficient Cdfus can be estimated by the use of the values contained in the above figures, obtained
from wind tunnel tests at MIT. All of the fuselages were 43 inches long. Note that other combinations
can be derived from these results. For instance, fuselages 1 and 8 are the same except that number 8
has landing gear. The increase in the coefficient is thus due to the landing gear.
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Resistance Models The program requires values for battery, ESC and motor resistances. Models were developed for each
of these resistances based on available information.
Battery Resistance
Battery resistance is a function of the number of cells in series (Ncells), Ampere hour capacity (AH) and
Discharge rating (C ). The general form of the equation is:
Rbattery =𝑎∗𝑁𝑐𝑒𝑙𝑙𝑠
𝐴𝐻∗𝐶+ 𝑏
For new batteries the value of a is about 0.52 and b, an allowance for wire and cables, has a value of
about 0.005 ohms. A value of 35 is chosen for C. It is assumed that a full power flight time of 6 minutes
or 0.1 hours is typical. Since Time = AH/Ibattery, AH = T * Ibattery or AH = 0.1 * Ibattery. In the program
it is assumed that each cell has a voltage of 4 volts, so Ncells = Ebattery/4.
Substituting:
Rbattery = .037∗𝐸𝑏𝑎𝑡𝑡𝑒𝑟𝑦
𝐼𝑏𝑎𝑡𝑡𝑒𝑟𝑦 +.005 Ohms
ESC Resistance
ESC resistance is usually very small but it is included for completeness.
Resc = . 281 ∗ 𝐼𝑏𝑎𝑡𝑡𝑒𝑟𝑦−1.24 Ohms
Motor Resistance
Rmotor = 6.9 ∗ 𝑃𝑏𝑎𝑡𝑡𝑒𝑟𝑦−0.842 +.005 Ohms
The motor resistance formula was derived by plotting the relationship between resistance and motor
power for a large number of motors. There is considerable scatter in the results so this formula may be
in error compared to a selected motor with known specifications. Nevertheless it provides a means of
estimating an approximate value for Kv. The added value of 0.005 is to account for connectors and
wires.